We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N, decide whether N>0. We show that PosSLP lies in the counting hierarchy, and combining our results with work of Tiwari, we show that the Euclidean Traveling Salesman Problem lies in the counting hierarchy -- the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.

We study two quite different approaches to understanding the complexityof fundamental problems in numerical analysis. We show that both hingeon the question of understanding the complexity of the following problem,which we call PosSLP: Given a division-free straight-line program producing an integer N, decide whether N>0.We show that PosSLP lies in the counting hierarchy, and we showthat if A is any language in the Boolean part of Polynomial-timeover the Reals accepted by a machine whose machine constants arealgebraic real numbers, then A is in P^PosSLP. Combining our resultswith work of Tiwari, we show that the Euclidean Traveling SalesmanProblem lies in the counting hierarchy -- the previous best upperbound for this important problem (in terms of classical complexityclasses) being PSPACE.

We thank Klaus Meer for calling our attention to the fact that Theorem 1.2follows very easily from Lemma 4.4 and Theorem 4.5 of the paper ``Saturation and Stability in the Theory of Computation over the Reals'', by Olivier Chapuis and Pascal Koiran (Annals of Pure and Applied Logic99 (1999) pp. 1-49). Their work shows more generally how algebraicconstants can be eliminated in computation over the reals.